Calculating Compound Formula from Crystal Structure
This question asks for the simplest chemical formula of a compound based on its cubic crystal structure. We need to determine the number of atoms of type $A$ and type $B$ within one unit cell.
Atom A Contribution
Atoms $A$ are located at the corners of the cubic unit cell.
A cubic unit cell has 8 corners. Each atom at a corner is shared by 8 adjacent unit cells. Therefore, the effective number of atoms $A$ contributed by the corners to a single unit cell is:
Number of $A$ atoms = (Number of corners) $\times$ (Contribution per corner)
Number of $A$ atoms = $8 \times \frac{1}{8} = 1$
Atom B Contribution
Atoms $B$ are located at the body centres of the cubic unit cell.
A cubic unit cell has 1 body centre. The atom at the body centre belongs entirely to that unit cell. Therefore, the effective number of atoms $B$ contributed by the body centre is:
Number of $B$ atoms = (Number of body centres) $\times$ (Contribution per body centre)
Number of $B$ atoms = $1 \times 1 = 1$
Determining Simplest Formula
The ratio of atoms $A$ to atoms $B$ in the unit cell is $1:1$.
Thus, the simplest formula of the compound is $AB$.
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